"WM" <mueckenh@rz.fh-augsburg.de> wrote in message news:195f6374-07c4-43cf-9416-3ca841f78cd3@fe28g2000vbb.googlegroups.com...

>Matheology § 220>>PA {{Peano-Arithmetik}} already tells us that the universe is>infinite, but PA stops after we have all the natural numbers. {{No,>Peano arithmetics never stops because it never reaches an end. Here>potential and actual infinity are confused.}} ZFC goes beyond the>natural numbers; in ZFC we can distinguish different infinite>cardinalities such as countable and uncountable, and we can show>that there are infinitely many cardinalities, uncountably many, etc.>{{and we can show that there is nothing of that kind other than in>dreams, but not in logic.}}>[Saharon Shelah: "Logical Dreams" (2002)]>http://arxiv.org/PS_cache/math/pdf/0211/0211398v1.pdf>>Regards, WM

this stuff is so stale (2002), unreviewed, and you failed to post the rest:

"But there are also set theories stronger than ZFC, which are as high above ZFCas ZFC is above PA, and even higher.1.4 The Scale Thesis: Even if you feel ZFC assumes too much or too little (andyou do not work artificially), you will end up somewhere along this scale, goingfrom PA to the large cardinals.(What does artificial mean? For Example, there are 17 strongly inaccessible2cardinals, the theory ZFC + there are 84 strongly inaccessible cardinals is con-tradictory and the theory ZFC + there are 49 strongly inaccessible cardinals isconsistent but has no well found model.)An extreme skeptic goes below PA, e.g., (s)he may doubt not only whether 2n(for every natural number n) necessarily exists but even whether n[log n] exists. [Inthe latter case (s)he still has a chance to prove there are infinitely many primes.]The difference between two such positions will be just where they put their belief;so the theory is quite translatable, just a matter of stress. For instance, by one weknow that there are infinitely many primes, by the other we have an implication.There is a body of work supporting this, the so called equi-consistency results (e.g.,on real valued measurable cardinals, see later).So far I have mainly defended accepting ZFC, as for believing in more, see later.6.12 Dream: Find natural properties of logics and nontrivial implications be-tween them (giving a substantial mathematical theory, of course).6.13 Dream: Find a new logic with good model theory (like compactness, com-pleteness theorem, interpolation and those from 6.12) and strong expressive powerpreferably concerning other parts of mathematics (see [Sh 702], possibly specificallyderive for them).6.18 Dream: Try to formalize and really say something4 on mathematical beautyand depth. Of course (length of proof)/(length of theorem) is in the right direction,etc.6.19 Dream: Make a reasonable mathematical theory when we restrict ourselvesto the natural numbers up to n, where n is a specific natural number (say 22100+1)(e.g., thinking our universe is discrete with this size)."